(25 points) A construction crew is lengthening a road. Let y represent the total length of the road (in miles). Let x represent
1 answer:
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Answer:
See explanation.
General Formulas and Concepts:
<u>Pre-Algebra</u>
Order of Operations: BPEMDAS
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
- Left to Right
<u>Algebra I</u>
Functions
- Function Notation
- Exponential Property [Rewrite]:
- Exponential Property [Root Rewrite]:
<u>Calculus</u>
Differentiation
- Derivatives
- Derivative Notation
Derivative Property [Multiplied Constant]:
Derivative Property [Addition/Subtraction]:
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Derivative Rule [Chain Rule]:
Step-by-step explanation:
We are given the following and are trying to find the second derivative at <em>x</em> = 2:
We can differentiate the 1st derivative to obtain the 2nd derivative. Let's start by rewriting the 1st derivative:
When we differentiate this, we must follow the Chain Rule:
Use the Basic Power Rule:
We know that y' is the notation for the 1st derivative. Substitute in the 1st derivative equation:
Simplifying it, we have:
We can rewrite the 2nd derivative using exponential rules:
To evaluate the 2nd derivative at <em>x</em> = 2, simply substitute in <em>x</em> = 2 and the value f(2) = 2 into it:
When we evaluate this using order of operations, we should obtain our answer:
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation